Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring $k[X,Y,Z]$ over a field k of characteristic zero extend to the polynomial ring $R[X,Y,Z]$ over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on $k[X_1,X_2,X_3,X_4]$ of rank at most three is a polynomial ring over k.     

Transcendence degree one function fields over a finite field with many automorphisms

Let $\mathbb{K}$ be the algebraic closure of a finite field ${\mathbb{F}}_q$ of odd characteristic p. For a positive integer m prime to p, let $F=\mathbb{K}(x,y)$ be the transcendence degree 1 function field defined by $y^q+y=x^m+x^{?m}$. Let $t=x^{m(q?1)}$ and $H=\mathbb{K}(t)$. The extension $F|H$ is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus $\mathfrak{g}(K)=(qm?1)(q?1)$, p-rank (Hasse–Witt invariant) $\gamma (K)={(q?1)}^2$ and a $\mathbb{K}$-automorphism group of order at least $2q^{2}m(q?1)$. In this paper we prove that this subgroup is the full $\mathbb{K}$-automorphism group of K; more precisely ${\text{Aut}}_{\mathbb{K}}(K)=\mathrm{\Delta }?D$ where Δ is an elementary abelian p-group of order $q^2$ and D has an index 2 cyclic subgroup of order $m(q?1)$. In particular, $\sqrt{m}|{\text{Aut}}_{\mathbb{K}}(K)|>\mathfrak{g}{(K)}^{3/2}$, and if K is ordinary (i.e. $\mathfrak{g}(K)=\gamma (K)$) then $|{\text{Aut}}_{\mathbb{K}}(K)|>{\mathfrak{g}}^{3/2}$. On the other hand, if G is a solvable subgroup of the $\mathbb{K}$-automorphism group of an ordinary, transcendence degree 1 function field L of genus $\mathfrak{g}(L)\ge 2$ defined over $\mathbb{K}$, then $|{\text{Aut}}_{\mathbb{K}}(K)|\le 34{(\mathfrak{g}(L)+1)}^{3/2}<68\sqrt{2}\mathfrak{g}{(L)}^{3/2}$; see [15]. This shows that K hits this bound up to the constant $68\sqrt{2}$.Since ${\text{Aut}}_{\mathbb{K}}(K)$ has several subgroups, the fixed subfield $F^N$ of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in ${\text{Aut}}_{\mathbb{K}}(K)$ is large enough. This possibility is worked out for subgroups of Δ.     

Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, $q=p^e$, and ${\mathbb{F}}_q$ be the finite field of q elements. Let $f,g\in {\mathbb{F}}_q[X,Y]$. The graph $G_q(f,g)$ is a bipartite graph with vertex partitions $P={\mathbb{F}}_q^3$ and $L={\mathbb{F}}_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l]=[l_1,l_2,l_3]\in L$ if and only if $p_2+l_2=f(p_1,l_1)$ and $p_3+l_3=g(p_1,l_1)$. If $f=XY$ and $g=X{Y}^2$, the graph $G_q(XY,X{Y}^2)$ contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs $G_q(f,g)$ containing no cycles of length less than eight and not isomorphic to the graph $G_q(XY,X{Y}^2)$, even without requiring them to be edge-transitive. So far, no such graphs $G_q(f,g)$ have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture.     

Reconstruction and subgaussian processes

This Note presents a randomized method to approximate any vector v from some set $T\subset {\mathbb{R}}^n$. The data one is given is the set T, and k scalar products ${(〈X_i,v〉)}_{i=1}^k$, where ${(X_i)}_{i=1}^k$ are i.i.d. isotropic subgaussian random vectors in ${\mathbb{R}}^n$, and $k\ll n$. We show that with high probability any $y\in T$ for which ${(〈X_i,y〉)}_{i=1}^k$ is close to the data vector ${(〈X_i,v〉)}_{i=1}^k$ will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. This extends and improves recent results by Candes and Tao. To cite this article: S. Mendelson et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.     

The second and fourth moments of theta functions at their central point

Let χ range over the $(p?1)/2$ even Dirichlet characters mod $p?3$, a prime. Let $\theta (x,\chi )$ be the associated theta series. It is known that the square mean value of $\theta (1,\chi )$ is asymptotic to $p^{3/2}/4\sqrt{2}$ as p goes to infinity. We prove that the fourth mean value of $\theta (1,\chi )$ is asymptotic to $\frac{3}{16\pi }{p}^2\log p$ as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p.     

Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.